A Remark on Bounded Homomorphisms into B ( H ) and Amenability of Unitary Groups

Let A be a unital amenable A∗-algebra, or a unital commutative hermitian Banach ∗-algebra. We show that if ρ : A → B(H) is a bounded homomorphism, then ρ is completely bounded and ‖ρ‖cb ≤ ‖ρ‖2. Mathematics Subject Classification: 46L07; 46k05


Introduction and Preliminaries
For a topological group G, LUC(G) is the space of left uniformly continuous bounded functions f : G → C. It is well known and easy to show that if f : G → C is bounded, then f ∈ LUC(G) if and only if the map x → x f is norm continuous from G to ∞ (G), where x f (y) = f (yx)(y ∈ G).The space LUC(G) is a unital C * -subalgebra of l ∞ (G), and is right invariant in the sense that x f ∈ RUC(G) whenever f ∈ RUC(G).If X is a right invariant, unital subspace of ∞ (G), then an element m ∈ X * is called a right invariant mean if m(1) = 1 = m and m( x f ) = m(f ) for all f ∈ X, and x ∈ G.We denote by R(X) the set of right invariant means on X.
Supposed that A is a C * −algebra.We denote by M n (A) the C * −algebra of n × n matrices with entries in A. For C * −algebra A and B, a (linear) map Ψ : A → B is called: positive if Ψ(a) ≥ 0 for every a ≥ 0; completely positive if Ψ (n) : M n (A) → M n (B) is positive for every n ∈ N, where Ψ We start by the following Theorem [10, Theorem 9.7]: Theorem 1.Let A be a commutative unital C * -algebra, and if ρ : In the proof this Theorem actually used from amenability of the unitary groups of C * -algebra A. The relationship between amenability of Banach algebras and amenability of groups (semigroups) is well known and consider.Johnson [6,Theorem 2.5] proved the remarkable result that if G is a locally compact group then G is amenable if and only if L 1 (G) is amenable Banach algebra.Paterson [9,Theorem 2] shows that a C * -algebra A is amenable if and only if U(A), the unitary group of A, is amenable.In the other hand we have the following example.

Example 1.
Let A be the disc algebra, we define an involution f → f * on A by f * (z) = f (z), then A is a Banach * -algebra.It is well known that A is not amenable.Now if S is the isometry semigroup of A, then S is a semitopological semigroup w.r.t. the relative (Banach space) weak topology since S is abelian thus is amenable, i.e.R(LUC(S)) = ∅.
We have attempted to develop Theorem 1, for unital amenable A * -algebras, and for unital commutative hermitian Banach * -algebras.The importance of the amenability (or the existence of an invariant mean) is best illustrated in the following Theorem.Theorem 2. (Dixmier) Let G be an amenable locally compact group, and let ρ : G → B(H) be a strongly continuous homomorphism with ρ(e) = 1, such that ρ u := sup{ ρ(t) : t ∈ G} is finite.Then there exists an invertible S in B(H) with S .S −1 ≤ ρ 2 u , such that S −1 ρ(.)S is a unitary representation of G.
The proof of this theorem [10,Theorem 9.3] holds also for the following version: Theorem 3. Let G be an amenable topological group, i.e., R(LUC(G)) = ∅ and let ρ : G → B(H) be a strongly continuous homomorphism with ρ(e) = 1, such that ρ u := sup{ ρ(t) : t ∈ G} is finite.Then there exists an invertible S in B(H) with S .S −1 ≤ ρ 2 u , such that S −1 ρ(.)S is a unitary representation of G.

Main results
First of all we state the following proposition.

Proposition 1. Let A be a unital Banach * -algebra such that the unitary group U(A) is a bounded set. Then A is amenable if and only if
Proof.Since U(A) is a bounded set, by [4,Theorem 34.3],A admits an equivalent C * -norm .* .So amenability of (A, .) is equivalent to amenability of (A, .* ) and by [9,Theorem 2] is equivalent to amenability of U(A).
Example 2. Let J be an arbitrary set of elements and consider the space of those complex-valued functions a(i, j) defined on J × J which satisfy the condition i,j |a(i, j)| 2 ≤ ∞.We make this set into an H * -algebra by the following definitions: if a = a(i, j), b = b(i, j), and λ is any complex number then It is easy to verify that with these definitions this set becomes an H * -algebra.
If n is the cardinal number of J then this algebra is called the full matrix H *algebra of order n, or sometimes simply a full matrix algebra.This algebra is amenable since its unitary group is a bounded set.Therefore every simple H * -algebra is amenable.
We recall that a Banach * -algebra (A, .) is said to be an A * -algebra provided there exists on A a second norm .* not necessarily complete, which is a C * -norm.The second norm will be called an auxiliary norm.The completion of (A, .* ) is a C * -algebra and is denoted by C * (A).Any C * -algebra is an A * -algebra.An important example of a Banach * -algebra whose norm does not satisfy the condition x 2 = x * x , but is an A * -algebra, is the group algebra L 1 (G) of a locally compact group G. Another example of an A * -algebra is provided by any semisimple real commutative Banach algebra.In this case, the involution is the identity map and auxiliary norm is the spectral radius r.Also, a semisimple hermitian Banach * -algebra is an A * -algebra, [3, Section 41, Corollary 10].Gelfand and Naimark [5] give an example of an A * -algebra, which is not hermitian.We denote by A sa the self-adjoint part of A.
If B is a unital C * -algebra with unitary group U(B), we consider on U(B) the relative weak topology (as a subset of the Banach space B).Regarding B ⊂ B * * , B * * is a von Neumann algebra.Give on the unitary group U(B * * ), the relative ultraweak topology (σ(B * * , B * )-topology).Now on U(B * * ) this topology coincides with both the weak operator and the strong operator toplogy on U(B * * ).Since involution is weak operator continuous and multiplication is strong operator continuous on U(B * * ), it follows that U(B * * ) is a topological group.Further since the weak topology on B coincides with the relative ultraweak topology, it follows that the topology on U(B) is the relative topology inherited from U(B * * ).Hence, U(B) is a topological group.

Corollary 2. Let A be a unital
where L 1 (H) is the set of all trace class operators on a separable Hilbert space H, with the trace norm .tr , and auxiliary norm given by the operator norm.Then, A is a A * -algebra and We recall that the involution in a * -algebra A is said to be hermitian if every hermitian element in A has real spectrum.When the involution in a * -algebra A is hermitian then A is called hermitian algebra.Many * -algebras are hermitian, for instance all C * -algebras are hermitian, as are the group algebras of abelian or compact groups.First we state the following Remark.

Remark 1.
Let A a unital hermitian Banach * -algebra, with an isometric involution.If ρ : A → B(H) is a * -representation, then the operator norm ρ ≤ 1: Clearly ρ maps unitaries to unitaries.Hence, ρ(u) ≤ 1 for any unitary u.Let h = h * be a self-adjoint element of A, with h ≤ 1, then 1−h 2 ≥ 0. By [1, Theorem 6.1.4.] there exists a self adjoint element is easily seen to be unitary.This shows that every self-adjoint element in the unit ball is the real part of a unitary.We obtain that ρ(h) ≤ 1 for every self-adjoint element h in the unit ball A. Since ρ(x) 2 Thus we have the following proposition.

Proposition 4.
Let A be a unital commutative hermitian Banach * -algebra, with an isometric involution.If ρ : A → B(H) is a bounded homomorphism, then ρ is completely bounded and ρ cb ≤ ρ 2 u .Proof.By (2.1) above, we have h = 1/2(u + u * ).This shows that every selfadjoint element is in the span of two unitaries.Using the Cartesian decomposition, we obtain that every element in A is a linear combination of at most four unitaries.let G denote the unitary group of A. Then, by (Dixmier's Theorem) Theorem 2, there is a similarity S, with S .S −1 ≤ ρ 2 , such that ρ(t) := S −1 ρ(t)S is unitary for all t in G. Since ρ(t * ) = ρ(t −1 ) = ρ(t) −1 = ρ(t) * (t ∈ G), thus ρ is self-adjoint on the unitary elements of A. Since every element in A is a linear combination of at most four unitaries, therefore ρ is a * -homomorphism on A.
In view of the obtained results, the following question is natural: Does the assertion of Corollary 2 holds for any unital hermitian Banach * -algebra?

Proposition 2 .Corollary 1 .
Let A be a unital A * -algebra with the auxiliary norm .* , G = U(A) the unitary group of A and B = C * (A).If we give G, the relative weak topology (as a subset of the Banach space B), then G is a topological group and is dense in U(B) w.r.t.this topology.Proof.G as a subgroup of topological groupU(B) is a topological group [2, Proposition 1.3.4].Suppose that x ∈ U (B) where U (B)) = {u ∈ U(B) : 1−u * < 2} then there exists a y ∈ B sa such that x = e iy [7, Theorem 2.1.12].Since A sa is .* -dense in B sa therefore there exists (y n ) ⊂ A sa such that y n → y in .* -topology, so (y n ) is .* -bounded (say by M), thus σ B (y n ) ⊂ [−M, M].Define f : [−M, M] → C by f (t) = e it , we know that f ∈ C([−M, M]) hence f (y n ) → f (y)or e iyn → e iy = x, in .* -topology.Note that there are two exponential functions here for e iy , one for each norm , but since .* is .-continuous they are agree.As y n ∈ A sa thus e iyn is unitary element of A. Consequently .* -closure of U(A) contains U (B).By [11, Theorem 4.11] U (B) is dense in U(B), w.r.t. the relative (Banach space) weak topology, so U(B) = w-closure of U(A).Let A be a unital * -subalgebra of a unital C * -algebra B. then G = U(A) the unitary group of A is a topological group and is dense in U(B) w.r.t. the relative weak topology.The following Corollary is a generalization of [9, Theorem 2].
[8,hermitian since a * -ideal (and hence spectral * -subalgebra) of the C * -algebra B(H)[8, Section 10.5].We know that A is not amenable because, it is well known that L 1 (H) does not admit a bounded approximate identity.But C* (A) = K(H) ⊕ C1 is amenable, therefore, R(LUC(U(A))) = ∅.Let A a unital amenable A * -algebra.If ρ : A → B(H)is a bounded homomorphism, then ρ is completely bounded and ρ cb ≤ ρ 2 u .Proof.Since A is a dense subalgebra of C * (A), thus closed unit ball of A is .* -dense in closed unit ball of B = C * (A), therefore .* -bounded sets and .-bounded sets in A are equals.Consequently ρ : A → B(H) is a .* -bounded homomorphism.We may ρ extend to a bounded linear map ρ : C * (A) → B(H). Cearly ρ is a homomorphism, and by [10, Theorem 9.7] ρ is completely bounded and ρ cb ≤ ρ 2 .By Proposition 2 U(C * (A)) = wclosure of U(A), and since ρ is w-w-continuous, therefore ρ u := sup{ ρ(t) :